I have the schedule below for 9 rounds which might work - just leave out one of the rounds. All the partnerships should be different and players should play against all others at least once and at most twice.
(10 5 v 11 15) (7 6 v 14 9) ( 8 12 v 16 1) ( 2 3 v 13 4)
(11 6 v 12 13) (8 4 v 15 7) ( 9 10 v 16 2) ( 3 1 v 14 5)
(12 4 v 10 14) (9 5 v 13 8) ( 7 11 v 16 3) ( 1 2 v 15 6)
(14 7 v 15 5) (9 11 v 6 10) ( 3 8 v 1 4) (16 12 v 2 13)
(15 8 v 13 6) (7 12 v 4 11) ( 1 9 v 2 5) (16 10 v 3 14)
(13 9 v 14 4) (8 10 v 5 12) ( 2 7 v 3 6) (16 11 v 1 15)
( 8 6 v 14 16) (9 7 v 1 12) (15 13 v 10 3) ( 4 2 v 5 11)
( 9 4 v 15 16) (7 8 v 2 10) (13 14 v 11 1) ( 5 3 v 6 12)
( 7 5 v 13 16) (8 9 v 3 11) (14 15 v 12 2) ( 6 1 v 4 10)
It is best to leave out one of the 1st three rounds.
